CALIBRATION OF RAILWAY BALLAST DEM MODEL

Ákos Orosz, János P. Rádics, Kornél Tamás

Department of Machine and Product Design

Budapest University of Technology and Economics

Műegyetem rkp. 3., H-1111, Budapest, Hungary

E-mail: orosz.aakos@gmail.com

KEYWORDS

Railway ballast, DEM, Calibration, Hummel device

ABSTRACT

The ballast of the railway track is constantly changing

due to dynamic forces of train traffic that results in the

crushing of the rocks. The feasibility to simulate each

particle makes the discrete element method (DEM)

suitable for the task.

Different DEM particle models are introduced in our

paper and the adequate one was chosen. This new method

is based on crushable convex polyhedral elements and

random shape generation via Voronoi tessellation and is

implemented in the Yade DEM simulation software.

The particle geometry is validated via comparing the

simplified shape of natural rough rocks and the randomly

generated ones. A 3D scanner was used to digitize the

natural rocks. The crushing behaviour is tested as well.

The validation of interaction laws and the calibration of

the micro parameters is necessary to create a DEM

material model with a realistic behaviour. In the

calibration process Hummel device is modelled, which

provides well measurable parameters for comparing

simulation and measurement results.

INTRODUCTION

In a limited number of cases, it is possible to model the

railway track ballast as a continuum with the use of finite

element method (Shahraki et al. 2015), however this

approach does not give information about many aspects.

Therefore a new approach should be utilised to simulate

the railway track ballast behaviour more realistically.

In our research, the rocks of the ballast are classified into

two groups based on their geometry: equant (Figure 1,a)

and flat (Figure 1,b). It is well known that both of the

shapes mentioned above have to be represented in the

railway track ballast to endure the forces effectively.

Moreover they have an optimal ratio, which is currently

estimated by routine of practice. The determination of its

exact value would provide many benefits.

0a) b)

Figure 1: a) Equant and b) Flat Rocks

(Asahina and Taylor 2011)

Result of the dynamic forces of the periodical loads is the

fragmentation of the rocks, which changes the ratio of the

different geometry types. This has a great effect on the

loadability of the aggregate. That is one of the reasons to

perform the complete maintenance of the railway track.

A main objective of our research is to investigate this

phenomenon and to determine the period between

maintenances more precisely through simulating the long

term behaviour of the ballast.

The full maintenance process has different stages

completed with different machines. The appropriate

model and simulation of the railway track ballast can be

used to improve efficiency of finding the optimal

machine settings and tool geometries.

Instead of the continuum approach, there is a need to

simulate each rock to achieve a realistic simulation,

whereby the previous questions can be answered. It is

also important to take efforts in the modelling of rock

breakage, because crushing of particles is necessary to

bring simulation data closer to experimental data

(Eliaš 2014). This makes the discrete element method

suitable for the task.

This paper represents a brief introduction of the discrete

element method, choosing of a discrete element material

model based on particle geometry and crushability

aspects, the mechanical basics of this model, a solution

for random geometry generation and validation of this

technique. Also an approach is introduced for calibrating

the model with the device used for the Hummel

measurement.

Proceedings 31st European Conference on Modelling and Simulation ©ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors) ISBN: 978-0-9932440-4-9/ ISBN: 978-0-9932440-5-6 (CD)

The Discrete Element Method

The discrete element method is a numerical technique,

where the simulated material is made of particles (or

elements), with independent motional degrees of

freedom. Interactions can be created and erased between

the elements (Bagi 2007). For this reason the behaviour

of the volume modelled with discrete element method

depends on the definition of the elements and interaction

laws. The sum of these features is called the discrete

element material model (or micromechanical properties).

The micromechanical properties have direct effect on the

element-level behaviour, which results in the measurable,

macromechanical behaviour. The aim of the study is to

reproduce the measurable characteristics of the aggregate

which can be achieved with a proper material model.

The exact relation between micro- and macromechanical

behaviour is missing in most cases, and the reasonable

parameters can only be reached by the calibration of the

material model. This is done by comparing the proper

measurement with its simulation, and modifying the

properties of the material model until its

macromechanical behaviour reproduces the test with

acceptable approximation. In our research the Hummel

measurement device is used to calibrate the material

model.

MATERIAL MODEL

The first difficulty of creating the model was the decision

about the shape of the particles. The simplest solution is

to create a volume made of spheres. This also gives an

advantage in computational speed. However it is known

that many type of aggregates such as sand – or in our case

the railway track ballast –cannot be modelled properly

with spheres because of the rolling contact surfaces

versus the actual friction sliding of real particle surfaces

(Lane et al. 2010).

To keep the advantages of spheres but to also be able to

use model elements with more complex shape, the so

called clumps are used. The clumps are made of spheres

with rigid connections between them. Example is shown

on Figure 2,a. (Coetzee and Nel 2014).

The clump particle is still smooth and is missing the sharp

corners and edges. Particle approximation can be

improved by reducing the element size of the clump

particle, but the highly increasing computation time have

to be accepted. Therefore an extensive effort was made

to simulate the aggregate with polyhedral elements. An

example of the shape of these particles is shown in

Figure 2,b .

There are different programs e.g. Grains3D (Wachs et al.

2012), BLOKS3D (Huang and Tutumler 2011) for

handling polyhedral particles, but these are hardly

accessible softwares and there is no opportunity to make

modifications. Jan Eliaš also created a polyhedral particle

model (Eliaš 2014), which features the possibility of

crushing particles with low computation need and is

implemented in the Yade open source DEM software.

The available source code creates the chance to make

modifications and improvements in the embedded

material models, which is a significant benefit. The built-

in crushing effect and the possibility of improvements are

the main reasons that the crushable polyhedral material

model (Eliaš 2014) was selected for further studies.

a) b)

Figure 2: Examples for Modelling Rocks with Clumps

(Coetzee and Nel 2014), and Polyhedra (Huang and

Tutumler 2011)

Creating polyhedral shaped elements

There are several ways to create the shape of the

polyhedra. The simplest way is to create them manually,

estimating the geometry of rocks with the desired

precision. Advanced way is using a 3D scanner and

simplifications (such method is described later in the

validation section), or even automatized image

processing technology has promising results (Huang and

Tutumler 2011). However, what is common in these

methods is that only a limited number of rocks can be

processed and then reproduced. There is a need to extend

the productivity of the creation process and to raise the

variability of the particles. There is no pattern in the

shape of the natural particles. Therefore it is feasible to

generate the shapes randomly without remarkable

deviation from the mined rocks. Such a solution is

implemented in the chosen material model, which was

inspired by Asahina and Bolander (2011) and relies on

the Voronoi method (Figure 3). In 2D it is based on

random points created in an area with a defined minimal

distance. Then polygons are created by finding the union

of apothems of the lines between the closest nuclei pairs.

Figure 3: Voronoi Method (Eliaš 2014)

Interactions between the elements

The material of the elements is ideally rigid. The effect

of plasticity is modelled within the definition of

interactions (repulsive force). The model is non-

cohesive. Normal and shear force is included, which arise

when the polyhedrons come into contact.

The forces act in the centroid of mass of the overlapping

volume. The magnitude of normal force between two

particles is obtained from the magnitude of their

overlapping volume by multiplying it with the factor

named normal volumetric stiffness.

The direction of the normal force is perpendicular to the

plane fitted by the least square method on the intersecting

lines of the particle shells.

The shear force (friction) is proportional to the mutual

movements and rotations of the elements. Its maximum

value is regulated by the Coulomb friction model

(Equation 1, where |Fs| is the magnitude of shear force,

|Fn| stands for the magnitude of normal force, and φ is the

internal friction angle).

|𝑭𝒔| ≤ |𝑭𝒏|𝑡𝑎𝑛𝜑 (1)

Many material models do not include velocity based

damping such as the mentioned polyhedral material

model (Eliaš 2014) or the model by D’Addetta et al.

(2001) to dissipate kinetic energy, so it is worth

considering to use an artificial damping. This numeric

damping decreases the forces which increase the particle

velocities and vice versa by a factor λd: 0-1 (Šmilauer et

al. 2015).

Crushing behaviour

A further aim of the research is to simulate the

fragmentation process of the rocks in the railway ballast,

which leads to a great decrease in loadability. This can be

obtained with a proper crushing model. (Eliaš 2014) also

has the conclusion that the modelling of crushing is

necessary to get a realistic behaviour of the simulated

aggregate.

In the simulation crushing occurs, when the von Mises

stress in the polyhedral element exceeds the so-called

size-dependent strength (ft). Using the von Mises stress

is a simplification, as it is best applies to isotropic and

ductile metals. However, this simplicity is an advance in

discrete element modelling, as the aim is always to find

the simplest material model which reproduces the

aggregate (macro) behaviour with the proper

micromechanical parameters.

According to Lobo-Guerrero and Vallejo (2005) the

strength of the rock is dependent on its size. This

phenomena is implemented in the model and is

represented by Equation 2. Where f0 is a material

parameter and req is the radius of the sphere with same

volume as the polyhedra.

𝑓𝑡 =𝑓0𝑟𝑒𝑞

(2)

When crushing occurs, rocks are intentionally split into 4

pieces (Figure 4). The two mutually perpendicular

cutting planes are perpendicular to the plane defined by

σI and σIII principal axes and form angles of π/4 with the

planes defined by σI-σII and σIII-σII axes.

Figure 4: Crushing of Rocks (Eliaš 2014)

VALIDATION

The investigation of the properties of the polyhedral

material model (Eliaš 2014) lead to a decision to use it to

simulate the behaviour of railway ballast.

The first step of the process is the validation of the

material model to determine if it is appropriate for the

task. Two uncommon features of the selected material

model are the particle geometry generation with Voronoi

method and the crushing effect. The validation of these is

performed by examining the particles individually. If the

validation is successful, the next step is the calibration of

the material parameters with the use of particle

aggregate.

Particle Shape validation

To validate the randomly created polyhedral particle

shapes, the generated ones were compared with the

natural ones. A 3D laser scanner (NextEngine) was used

to digitize the natural rock shapes. This resulted in a very

complex surface geometry, which was simplified in a 3D

CAD software. Simple planes were fitted on the rough

surfaces using the least-square method (Figure 5). The

planes were trimmed by the penetration lines. Between

the manually created shapes and the randomly generated

ones a high level of similarity was well observable. The

average number of sides, vertices, as well as the areas of

the sides and magnitudes of angles was equal with a good

approximation.

A substantial simplification of the random generation is

that it always creates convex polyhedra. It is easy to find

concave areas on the surface of crushed rocks, which are

possible stress concentration spots. The magnitude of the

stress concentration effect depends on the size and shape

of these concave areas and has a significant influence on

the strength of the rocks.

Despite the existence of stress concentration effect at

concave polyhedra, convex ones are accepted. The

reason is that their strength is adjusted with the size-

dependent strength material parameter and their efficient

generation is also possible with Voronoi tessellation. The

geometry of the crushed rocks can be estimated by the

randomly generated ones relying on the Voronoi method.

Figure 5: Steps of Rock Shape Digitalizing with 3D

Scanner

Crushing test of a simple rock particle

An individual polyhedron has to crush into 4 pieces,

when the von Mises stress exceeds its size-dependent

strength. It occurs when the load on an element reaches a

critical magnitude. The crush can occur multiple times,

however, the fragmentation has to stop beside the same

load, as the created new elements have greater strength

because of their smaller size.

The following stages can be observed in the process

(Figure 6):

1. Beginning of the load: the loading plate has not

reached the rock yet.

2. The plate reaches the rock, the force starts rising.

3. Crush of the rock: when the von Mises stress exceeds

the strength of the rock, it breaks into 4 pieces. After the

crush, the normal force drops down to zero as the contact

between the rocks and the plate interrupts.

4. Moving, sliding of the rocks. When they reach their

final place, the force starts rising steeply again.

5. A second break occurs. In this case the force does not

reach zero, as the contact persists.

6. Reaching the maximum force and beginning of

unloading.

Figure 6: Loading Process of a Rock (Fy: Normal Force

[N], y: Horizontal Displacement of the Plate

Downwards [m])

The polyhedron breaks into 4 fragments at every crush as

expected. This is certainly not a natural behaviour,

however it is not an issue, because the research is more

intended to create the proper model of the whole

aggregate. Only the number of crushes and the further

behaviour of the aggregate is important.

Despite the fact that splitting intentionally into 4 pieces

is certainly not a natural behaviour, the model can be used

in further applications to analyse the behaviour of an

aggregate.

CALIBRATION OF THE MATERIAL MODEL

To create the model of the railway ballast with

approximately the same macromechanical parameters as

the natural aggregate, the proper setting of

micromechanical parameters is needed. It can be

obtained by calibration. During the calibration process,

the simulation of a properly chosen measurement is

compared to the corresponding experimental data. The

adequate measurement process applies similar loads as

the forces acting in natural aggregate, so the appropriate

material parameters can be set. In our case, the

measurement device used in the Hummel procedure is

applied for the calibration process, where crushing occurs

as the effect of quasi-static loads.

Hummel Device

The Hummel procedure (Hungarian Standards Institution

1983) gives information about the static loadability of

construction aggregates. The standard describes the

properties of the tested material, the geometry of the tool

(Figure 7) and the process of loading progress. Particles

of the aggregate are crushing during the test. The

Hummel process enables the determination of force-

displacement and particle size distribution curves under

different peak loads for the tested aggregate. Tested

material parameters have to be changed in order to be

able to perform the test, as the 31.5/50 mm railway ballast

rocks are too big for the Hummel device. Therefore, the

laboratory tests and the DEM simulation is performed

with 22/32 mm rocks. In the further research, grain size

sensitivity simulations and measurements will be

performed with smaller rocks in order to get information

about the particle size dependency of the aggregate

behaviour. The parameters of the 31.5/50 mm railway

ballast rocks can be set by extrapolation.

Figure 7: Hummel Device (Hungarian Standards

Institution 1983)

Creating the Geometry

To obtain the optimal processing speed besides the

proper geometry, the tube was approximated with a

regular decagonal prism, created from triangular facets.

The polyhedral particles were generated in this volume

with the desired geometry (equant or flat). In the first

stage of the research, equant elements were modelled.

The randomly generated polyhedra are created in a

cuboid volume (Figure 7/a.) inside the prism. To create a

dense pack of polyhedra on the bottom of the tube model,

gravitational deposition was used (Figure 8.). Gravity

load is applied in the simulation and the elements fall

down freely to the bottom of the prism. The height of the

produced dense pack is bigger than the Hummel device

has, so the elements with centroid higher than 150 mm

were removed (Figure 9).

a)

b)

Figure 8: a) Created Polyhedra; b) the Result of

Gravitational Deposition

a) b)

Figure 9: a) The Polyhedra Pack Before; b) and After

the Removal Process

Applying the Load

The initial and the maximum load state of the simulation

can be observed on Figure 10. The load is applied through

the top plate which is also a polyhedron type element

with disabled crushing and has a special shape to fit in

the tube. The plate moves down with constant velocity,

thus the magnitude of the force on the plate is

displacement driven, which is constantly measured.

When the pre-defined maximum force is reached, the

direction of the plate movement changes and the

unloading process begins. The simulation ends at the

moment, when the normal force reaches zero. Data are

saved in text format.

The force-displacement curve of a typical simulation is

represented on Figure 11. It corresponds to the

theoretical assumptions stated in the followings. In the

first part of the loading process the normal force raises

slowly. In this session the elements can move easily

because of the high porosity of the aggregate. As the

porosity decreases, the elements have less space to move,

the curve becomes steeper, and also the rate of crushing

events increases. In the unloading phase, the normal force

drops down to zero. Crushing can cause peak forces,

which eliminate in a few timesteps and have no

significant influence on the loading characteristics.

Therefore it is now assumed that they can be ignored, but

further investigations will be done.

Considering that the simulation used preliminary

material parameters, the results are acceptable and the

material model and calibration method can be utilized in

the further research.

a) b)

Figure 10: a) The Device Before Beginning of Load;

b) at Maximum Load

Figure 11: Normal Force (Fy [N])-Displacement (y [m])

Graph of the Hummel Device

CONCLUSIONS

In this paper an ongoing research for modelling the

railway ballast is introduced. Different DEM material

models were investigated, and the one featuring

randomly generated convex crushable polyhedral

elements was chosen. The mechanical basics of the

model, random element generation via Voronoi

tesselation and a crushing behaviour was investigated

and validated. A calibration method was created that uses

the Hummel device.

In the studied discrete element material model the

geometry of the polyhedral elements approximates the

shape of the rocks sufficiently, the mechanical model

reproduces the behaviour of the rock aggregate, and the

crushing works. The discrete element method is capable

to simulate the railway ballast.

The gravitational deposition executes with the assigned

micro parameters in the created geometry. The

characteristics of the load-displacement curve is the same

as theoretically expected. In some cases, peak forces

occur during breakage, but they have no significant effect

on the nature of the curve. In further studies they will be

investigated in detail.

The model of the measurement based on the Hummel

device is proper, so the calibration of the discrete element

material model can be performed in the further stages of

the research.

FURTHER RESEARCH

The next step is the static calibration of the material

model. It is performed by processing the measurement

data, and running a series of simulations of the Hummel

device measurement with different material parameters

to find the parameter combination that reproduces the

experimental data with the best approximation.

A calibrated and validated material model can be used to

simulate the behaviour of railway ballast and answer the

technical questions discussed in the introduction.

ACKNOWLEDGEMENT

This research is a part of an R&D project supported by

Magyar Államvasutak Zrt. The support is gratefully

acknowledged. We are thankful for the fellow workers of

the Department of Construction Materials and

Technologies, BME: Miklós P. Gálos professor for

coordinating the project and Gyula Emszt, department

engineer and Bálint Pálinkás technician for performing

the laboratory measurements.

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AUTHOR BIOGRAPHIES

ÁKOS OROSZ was born in Hódmezővásárhely,

Hungary. He is doing his mechanical engineering MSc

studies at the Budapest University of Technology and

Economics, where he received his BSc degree in 2016.

He is also a member of a research group in the field of

discrete element modelling. His e-mail address is:

orosz.aakos@gmail.com and his web-page can be found at http://gt3.bme.hu/oroszakos.

JÁNOS P. RÁDICS is an assistant professor at Budapest

University of Technology and Economics where he

received his MSc degree. He completed his PhD degree

at Szent István University, Gödöllő. His main research is

simulation of soil respiration after different tillage

methods, and he also takes part in the DEM simulation

research group of the department. His e-mail address is:

radics.janos@gt3.bme.hu and his web-page can be found at

http://gt3.bme.hu/radics.

KORNÉL TAMÁS is an assistant professor at Budapest

University of Technology and Economics where he

received his MSc degree and then completed his PhD

degree. His professional field is the modelling of granular

materials with the use of discrete element method

(DEM). His e-mail address is: tamas.kornel@gt3.bme.hu

and his web-page can be found at http://gt3.bme.hu/tamaskornel.